Problem: Factor the following expression: $9$ $x^2+$ $52$ $x+$ $35$
This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(9)}{(35)} &=& 315 \\ {a} + {b} &=& & & {52} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $315$ and add them together. The factors that add up to ${52}$ will be your ${a}$ and ${b}$ When ${a}$ is ${7}$ and ${b}$ is ${45}$ $ \begin{eqnarray} {ab} &=& ({7})({45}) &=& 315 \\ {a} + {b} &=& {7} + {45} &=& 52 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {9}x^2 +{7}x +{45}x +{35} $ Group the terms so that there is a common factor in each group: $ ({9}x^2 +{7}x) + ({45}x +{35}) $ Factor out the common factors: $ x(9x + 7) + 5(9x + 7) $ Notice how $(9x + 7)$ has become a common factor. Factor this out to find the answer. $(9x + 7)(x + 5)$